
In this lecture we'll discuss modal analysis at a relatively high level and explore conceptually how it can help us in determining the response of MDoF systems. Let's start by recalling the system of $N$ coupled equations of motion from the last section,

$$
\begin{bmatrix}
M
\end{bmatrix}
\begin{Bmatrix}
\ddot{u}
\end{Bmatrix} +
\begin{bmatrix}
C
\end{bmatrix}
\begin{Bmatrix}
\dot{u}
\end{Bmatrix} +
\begin{bmatrix}
K
\end{bmatrix}
\begin{Bmatrix}
u
\end{Bmatrix}=
\begin{Bmatrix}
P
\end{Bmatrix}
\tag{1}
$$

As a recap, we said that because these equations are coupled, the only practical way to solve them was by using direct numerical integration. The fact that they are coupled also means we need to solve them in parallel, simultaneously. The problem with this approach is that it leaves a lot of 'baked in' dynamic information about the structure hidden.

This is where **modal analysis** and the process of **modal superposition** come in. We're going to dig into the details in the following lectures but the basic idea of modal analysis is that we can perform a coordinate transformation using a special matrix, $[\Phi]$ known as the **modal matrix**,

$$
\begin{bmatrix}
\Phi
\end{bmatrix}^T
\begin{bmatrix}
M
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix}
\tag{2}
$$

$$
\begin{bmatrix}
\Phi
\end{bmatrix}^T
\begin{bmatrix}
C
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix}
\tag{3}
$$

$$
\begin{bmatrix}
\Phi
\end{bmatrix}^T
\begin{bmatrix}
K
\end{bmatrix}
\begin{bmatrix}
\Phi
\end{bmatrix}
\tag{4}
$$

This transformation has the effect of uncoupling the mass, damping and stiffness matrices. This will ultimately allow us to represent the system of $N$ coupled equations as an equivalent system of $N$ uncoupled equations.

This has huge practical value, as it allows us to evaluate the $N$ independent SDoF systems and superimpose their responses to determine the combined response of the system.

Now, there are some complications around damping that require further discussion. So for now, we'll remove damping from the discussion and only consider undamped vibration. Damping will make a return in the next section.

### Vibration modes and the modal matrix $[\Phi]$

The motion of a NDoF system can be considered as the summation of $N$ **modes of vibration**. We can make a simple analogy to the static analysis of a beam under the influence of 2 loads. The deflected shape of the structure under the action of both loads can be obtained as the summation of the deflected shapes obtained when both loads act individually. This is the simple principle of superposition and it maps across to dynamic modal analysis.

So, in a similar way, for a NDoF structure, the total vibration response can be obtained from the summation of responses in $N$ independent modes of vibration.

Remember that we are dealing with structural vibration so the response of a structure is a function of both time and space (position along the structure). So to characterise a vibration mode we need,

- a mode shape, telling us how the vibration response varies across the structure (variation in space)
- a modal frequency, telling us the frequency with which the structure oscillates in that shape or pattern (variation in time)

Because of the uncoupling that takes place as a result of the coordinate transformation discussed above, the oscillatory behaviour in each mode is independent.

To simulate the dynamic response of any structure using a modal decoupling approach, all we need to do is determine the modal frequencies and mode shapes (both of which are inherent properties of the structure), analyse the modal dynamic behaviour and then combine the modal responses in the process of modal superposition.

The modal matrix $[\Phi]$ is an $N\times N$ matrix whose columns are the mode shape vectors discussed previously. The mode shape vectors possess an important property called **orthogonality**. We'll soon see that it is this special property that facilitates the decoupling of the equations of motion.

### Analysis process overview

The process of modal analysis can be summarises as follows:

- For any NDoF system, determine $N$ modal frequencies and their corresponding $N\times 1$ mode shape vectors. Remember, the mode shape vectors represent the pattern of displacement in a given mode while the modal frequency is the natural frequency of the SDoF system that characterising the time-varying response in the mode.

- Use the mode shape vectors to construct the modal matrix $[\Phi]$.

- Use $[\Phi]$ to perform the transformation to normal/modal/generalised coordinates that decouples the equations of motion for the NDoF system.

- Solve for the response of $N$ independent SDoF systems, reverse the coordinate transformation and superimpose the responses to recover the complete geometric response of the structure.

### Advantages of a modal approach

The modal analysis approach has the following advantages over direct simultaneous solution performed in the previous section:

- It allows us to use the relatively simple structural dynamics of SDoF systems to solve for the behaviour of MDoF systems.

- Decomposing the response into $N$ vibration modes allows us to identify if the structure is particularly susceptible to dynamic loading with a specific frequency. For example, if we are designing a footbridge we might be particularly concerned if our modelling suggested the bridge has a modal frequency around $2 \:Hz$. Since the average human pacing frequency is around $2 \:Hz$, dynamic loads imposed by walking pedestrians would resonate with the $2 \:Hz$ vibration mode and could lead to a large dynamic response.

- Knowledge of the mode shapes allows us to identify regions of a structure that will be particularly susceptible to dynamic loading at the corresponding modal frequencies. We can expand on this further below.

Consider a simply supported beam, Fig 1. The beam is a continuous structure - for the purposes of this explanation discretisation is not important.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/what-is-modal-analysis-and-how-does-it-help-us/img1.jpg"
	alt="Simply supported beam subject to a dynamic load. | EngineeringSkills.com"
	width="400"
	height="163"	
/>
<Caption>
  Fig 1. Simply supported beam subject to a dynamic load $p(t) = \sin{\omega t}$ at $\alpha L$ from the left support.
</Caption>

Because the beam is continuous rather than consisting of $N$ discrete masses, it actually has an infinite number of vibration modes, although their significance decreases rapidly as the modal frequency increases. The modal frequencies can be obtained as,

$$
\omega_n = n^2\pi^2\sqrt{\frac{EI}{\bar{m}L^4}}
\tag{5}
$$

where $N$ is the mode number and $\bar{m}$ is the mass per unit length. The mode shape that corresponds with each frequency is given by,

$$
\phi_n(x) = \sin\frac{n\pi x}{L}
\tag{6}
$$

Now we can plot the first three modes and let's imagine that the beam has modal frequencies,

$$
\omega_1 = 1\:Hz
\tag{7}
$$

$$
\omega_2 = 4\:Hz
\tag{8}
$$

$$
\omega_3 = 9\:Hz
\tag{9}
$$

```python
L=1 #Beam length
x = np.arange(0,L+0.01,0.01) #Vector of x-coordinates
phi_1 = np.sin(math.pi*x/L) #Mode 1 shape
phi_2 = np.sin(2*math.pi*x/L) #Mode 2 shape
phi_3 = np.sin(3*math.pi*x/L) #Mode 3 shape

fig, axes = plt.subplots(figsize=(12,14),nrows=3,ncols=1)
axes[0].plot(x,phi_1,'-')
axes[0].plot([0,1],[0,0],'black')
axes[0].plot([0.5,0.5],[0,1],'red')
axes[0].set_xlim([0,1])
axes[0].set_ylabel('$\phi_1(x)$')
axes[0].set_title('Mode shape for mode 1')
axes[0].grid()

axes[1].plot(x,phi_2,'-')
axes[1].plot([0,1],[0,0],'black')
axes[1].plot([0.25,0.25],[0,1],'red')
axes[1].plot([0.75,0.75],[0,-1],'red')
axes[1].set_xlim([0,1])
axes[1].set_ylabel('$\phi_2(x)$')
axes[1].set_title('Mode shape for mode 2')
axes[1].grid()

axes[2].plot(x,phi_3,'-')
axes[2].plot([0,1],[0,0],'black')
axes[2].plot([1/6,1/6],[0,1],'red')
axes[2].plot([0.5,0.5],[0,-1],'red')
axes[2].plot([5/6,5/6],[0,1],'red')
axes[2].set_xlim([0,1])
axes[2].set_ylabel('$\phi_3(x)$')
axes[2].set_title('Mode shape for mode 3')
axes[2].grid()

plt.show()
```

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/what-is-modal-analysis-and-how-does-it-help-us/img2.jpg"
	alt="Mode shapes for modes 1, 2 and 3 of a simply supported beam. | EngineeringSkills.com"
	width="1000"
	height="1084"
	caption="Fig 2. Mode shapes for modes 1, 2 and 3 of a simply supported beam (top to bottom)."
/>

To highlight the influence of mode shape and frequency we can consider some scenarios; first, if $p(t)$ has a frequency of $1\:Hz$ and is positioned at $0.5\:L$,

- it will resonate with mode 1. Furthermore, it will have the maximum possible influence on mode 1 because it is positioned at the maximum position of mode shape 1. This combination of forcing frequency and position means mode one will dominate the overall structural response.

- it will have no influence on mode 2 because it is applied at a 'zero point' on mode shape 2. This means that the total response of the beam will have no contribution from mode 2.

- it will have an influence on mode 3 due to its position at a maximum in the mode shape, but the frequency ratio will mean the influence of mode 3 on the overall response is quite small in comparison to mode 1.

In the second scenario, if the frequency of $p(t)$ is increased to $5\:Hz$ and its position is shifted to say $0.4\:L$ it will excite all modes due to its position at a non-zero point in each mode shape, but the strength of excitation of each mode will be governed by the frequency ratio. As mode 2 has the closest frequency to the forcing frequency it will experience the largest response. It will therefore make the largest contribution to the structural response when the modal responses are superimposed.

From this discussion it should be clear that knowledge of the modal properties of a structure is very useful at the design stage or when trying to understand the dynamic behaviour of structures in service. This is why modal analysis and the modal superposition method has become the dominant method of analysis for MDoF systems. This is particularly so in the fields of civil and structural engineering.

One note of caution before we wrap up this lecture; modal analysis relies on superposition. As such, like any superposition technique it is limited to cases of linear behaviour. If it is expected that the response is non-linear, it may be appropriate to use modal analysis to determine the modal properties, but modal superposition should not be used to determine the structural response.


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